(Hidden) assumptions of simple compartmental ODE models Juliet Pulliam, PhD South African DST-NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA) Stellenbosch University MMED 2018 Slide Set Citation: DOI:10.6084/m9.figshare.5044606 The ICI3D Figshare Collection

Goals Review commonly overlooked assumptions that are inherent in the structure of simple compartmental ODE models Discuss when these assumptions might be problematic, and when they may be desirable Begin to explore some alternative model structures that relax the assumptions Should continuous treatment of individuals be recast as an assumption of large population size? maybe Should continuous treatment of time be recast as non-varying (constant) rates? Probably not

Taxonomy of compartmental models Continuous treatment of individuals (averages, proportions, or population densities) Discrete treatment of individuals Deterministic continuous time Ordinary differential equations Partial differential equations discrete time Difference equations (eg, Reed-Frost type models) Stochastic Yesterday, at the end of the day, I showed you what I referred to as a ‘taxonomy of compartmental models’ – before we move on, let’s first review what’s meant by a compartmental model… continuous time Stochastic differential equations discrete time Stochastic difference equations continuous time Gillespie algorithm discrete time Chain binomial type models (eg, Stochastic Reed-Frost models)

Taxonomy of compartmental models What is a compartmental model? A compartmental model is a model that partitions the population it represents into types and treats all individuals of a given type as identical; For example, yesterday I introduced the basic the SIR model The equations I showed you were ordinary differential equation implementations of the more general model structure represented here, which describes individuals as Susceptible, Infectious, or Recovered (and shows the processes that allow individuals to flow [into, out of, and] between these states) By writing the equations down as ODEs, I made a number of additional assumptions that weren’t explicitly discussed

Taxonomy of compartmental models Continuous treatment of individuals (averages, proportions, or population densities) Discrete treatment of individuals Deterministic continuous time Stochastic The first of these assumptions is the Continuous treatment of individuals This is really an assumption that the population is large enough that you can ignore the random fluctuations induced by the fact that individuals are discrete units and think about AVERAGE SYSTEM BEHAVIOR large population size continuous time

Demographic stochasticity time time time Borchering & McKinley (2018) Multiscale Modeling and Simulation DOI: 10.1137/17M1155259

Taxonomy of compartmental models Continuous treatment of individuals (averages, proportions, or population densities) Discrete treatment of individuals Deterministic continuous time Stochastic Compartmental ordinary differential equation models also assume that you can ignore any other source of random variation that might allow variability in the outcome large population size continuous time

Environmental stochasticity

Compartmental ODE models assume Large population size Deterministic progression for a given set of initial conditions and parameter values, a deterministic model always gives the same outcome Quickly assess what model outcomes are possible, and when These assumptions make compartmental differential equation models relatively easy to work with So one can relatively quickly determine what model outcomes are possible, and how the outcome is determined by the parameter values and the initial conditions. This is a really useful property – for example – when you are conducting a sensitivity analysis

Compartmental ODE models assume Large population size Deterministic progression for a given set of initial conditions and parameter values, a deterministic model always gives the same outcome Breaking Assumptions! Discrete Individuals and Finite Populations However, breaking these assumptions is often necessary – particularly when your research question focuses either on small populations, or on understanding variability in the possible outcomes (of a process or intervention) and Becky’s session this afternoon (in Track B) will cover several modeling approaches that will allow you to modify these assumptions

Taxonomy of compartmental models Continuous treatment of individuals (averages, proportions, or population densities) Discrete treatment of individuals Deterministic continuous time Ordinary differential equations N Stochastic The next several assumptions I’ll talk about that are inherent in the structure of simple compartmental ODE models relate to the way they treat time To illustrate these points, I’ll use an even simpler ODE model, with only one compartment (representing the population) and two processes, birth (by which individuals can move into the compartment, and death, by which individuals are removed from the compartment {the state variables are updated through time}

N Simple ODE models assume Time proceeds in a continuous manner Parameter values remain constant continuous time Ordinary differential equations N In this model… Time proceeds in a continuous manner, and The per capita birth and death rates do not change This means – for example – that the model explicitly assumes that whatever’s happening at 3am is what’s happening at 3pm, and whatever’s happening in summer and is exactly the same in winter For many procesess this is, of course, not a good representation of reality And again – depending on your research question – these assumptions may or may not make sense for your model. Luckily, breaking this assumption is relatively easy. This is what the differential equation version of this model looks like. It can be simplified further by writing b-d as r, a quantity …

N Simple ODE models assume Time proceeds in a continuous manner Parameter values remain constant continuous time Ordinary differential equations N This is what the differential equation version of this model looks like, and it can be simplified further by writing b-d as r, a quantity that’s known as the instantaneous intrinsic population growth rate This equation can now be solved analytically by integrating from some time t to a future time t + delta t

N continuous time Ordinary differential equations discrete time Difference equations N Doing this gives us a natural way move between a differential equation formulation of the model and a discrete-time formulation that allows you to represent processes that vary in time (such as seasonal birth and death processes) by averaging across temporally varying parameters for some specified delta t.

Simple compartmental ODE models assume Homogeneity within compartments Large population size Deterministic progression Time proceeds in a continuous manner Parameter values remain constant Memory-less processes So far we’ve discussed five assumptions that are inherent in the structure of simple compartmental ODE models (list) The final assumption I’ll discuss is the one that is perhaps the most often overlooked in infectious disease modelling, and one that can have important consequences for epidemic dynamics: The assumption of memory-less processes A memory-less process is one where the waiting time until an event occurs doesn’t depend on how long it’s been since the last event occurred Again, we’ll use an extremely simple model to illustrate the point – this time with a single compartment and a single process, by which individuals leave the compartment; this could be thought of as death

Simple ODE models assume Memory-less processes N Proportion surviving to at least τ τ Again, we can understand the implications of this assumption by integrating the differential equation that corresponds to our very simplified model, this time from the initial time 0 to an arbitrary time we’ll call tau We can now divide both sides by the initial population size to calculate the proportion of the population that survives to at least time tau Looking at the shape of this curve, you can see that under this assumption, most individuals will leave the compartment very quickly and a small proportion of the population will remain in the population for a very long time. The distribution of times until an event occurs is known as a waiting time distribution, and the WTD for an even that occurs at a constant per capita rate, such as in this example, is known as an exponential WTD, for reasons that should be obvious.

Realistic waiting times Pneumonic plague in India Mean is 4.5 days; shape is 2.3 For Gamma: 5.6 % > 10 days For exponential >11% > 10 days Estimate based on data from Joshi et al. (2009) Transactions of the Royal Society of Tropical Medicine and Hygiene

Simple compartmental ODE models assume Homogeneity within compartments Large population size Deterministic progression Time proceeds in a continuous manner Parameter values remain constant Memory-less processes We’ll talk more about this assumption and ways to break it tomorrow during the HIV in Harare tutorial. Non-exponential waiting times Breaking Assumptions!

Summary Simple ODE models are important tools for building understanding So, in summary … 1)It is possible to understand how all the parts interact to drive average system behaviour, especially for large populations For example, we can calculate analytically what proportion of the population needs to be vaccinated to prevent an epidemic and use this to develop an intuition for why some infections are easier to eradicate than others, as we saw yesterday

Summary Simple ODE models are important tools for building understanding It’s important to recognize the assumptions built into these models When populations are small, average behaviors can be misleading 2)

Summary Simple ODE models are important tools for building understanding It’s important to recognize the assumptions built into these models When populations are small, average behaviors can be misleading When rates vary, simple ODEs can fail to reproduce important (observed) dynamics Data from Earn et al. 2000 Science 2) through time or between individuals, or with the time spent in a compartment

Dushoff lecture on heterogeneity (Wed) Summary Simple ODE models are important tools for building understanding It’s important to recognize the assumptions built into these models When populations are small, average behaviors can be misleading When rates vary, simple ODEs can fail to reproduce important (observed) dynamics 2) through time or between individuals, or with the time spent in a compartment Dushoff lecture on heterogeneity (Wed)

Summary Simple ODE models are important tools for building understanding It’s important to recognize the assumptions built into these models When populations are small, average behaviors can be misleading When rates vary, simple ODEs can fail to reproduce important (observed) dynamics 2) through time or between individuals

Model Worlds A model world is an abstraction of the world that is simple and fully specified, which we construct to help us understand particular aspects of the real world A mathematical model is formal description of the assumptions that define a model world We know exactly what assumptions we’ve made, and we can follow those assumptions to their logical conclusions to address research questions

abstraction implementation REAL WORLD MODEL WORLD MODEL interpretation understanding

This presentation is made available through a Creative Commons Attribution license. Details of the license and permitted uses are available at http://creativecommons.org/licenses/by/3.0/ © 2014-2018 International Clinics on Infectious Disease Dynamics and Data Pulliam JRC. “(Hidden) Assumptions of Simple Compartmental ODE Models” Clinic on the Meaningful Modeling of Epidemiological Data. DOI: 10.6084/m9.figshare.5044606. For further information or modifiable slides please contact figshare@ici3d.org. See the entire ICI3D Figshare Collection. DOI: 10.6084/m9.figshare.c.3788224. 000 000 000